Put 1 Tbs. of the oil in a large saucepan or small soup pot, add the onion, and cook over medium heat until the onion has softened but not browned, about 3 minutes. Add the chili powder and tomato paste and stir with a wooden spoon to mix and cook briefly; take care not to let the chili powder scorch.
Season the chicken thighs lightly with salt and nestle them in the chile paste, turning them once so they’re entirely coated. Pour in about 1/2 cup of the broth and adjust the heat to a simmer. Cover the pan and cook the chicken, turning once, until it’s extremely tender when pierced with a knife, 30 to 40 minutes (add a little more broth if the pan is drying out). When the chicken’s done, remove it from the pan, let it cool a bit, and cut or shred it into bite-size pieces, discarding any bones and bits of fat or gristle; set aside.
If there’s any visible grease in the pan, spoon it off, add the remaining broth and the cilantro stems and simmer, uncovered, until the broth has reduced by about one-third and is quite flavorful, 20 to 30 minutes.
While the broth is reducing, fry the tortillas: Linke a plate with two layers of paper towels. Fill a small, high-sided saucepan with enough oil to come to about a 1-inch depth. Heat the oil over medium heat; when it reaches 375°F or when a strip of tortilla sizzles immediately when dipped in the oil, add six to eight strips of tortilla. With tongs or a long fork, "scrunch" them for a second or two so they take on a wavy shape. Fry until the strips aren't bubbling much and have become golden brown, about 1 minute. Transfer to the paper towels. Repeat with the remaining strips.
Divide the shredded chicken, the corn, black beans, tomato, and tortilla strips between two large soup or pasta bowls. Reheat the broth if necessary so it’s piping hot and pour it over the ingredients in the bowls. Serve immediately, and let each diner add the avocado, cheese, sour cream, chopped cilantro, and a big squeeze of lime juice at the table.